Skip to main content
×
×
Home

Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems

  • K. Manickam (a1) and P. Prakash (a1)
Abstract
Abstract

In this paper, a priori error estimates are derived for the mixed finite element discretization of optimal control problems governed by fourth order elliptic partial differential equations. The state and co-state are discretized by Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. The error estimates derived for the state variable as well as those for the control variable seem to be new. We illustrate with a numerical example to confirm our theoretical results.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems
      Available formats
      ×
Copyright
Corresponding author
*Corresponding author. Email addresses: pprakashmaths@gmail.com (P. Prakash), kkmmanickam@gmail.com (K. Manickam)
References
Hide All
[1] Adams R. A. and Fournier J. J. F., Sobolev Space, Academic Press, New York, 2008.
[2] Arada N., Casas E. and Tröltzsch F., Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl., 23 (2002), pp. 201229.
[3] Becker R., Estimating the control error in discretization PDE-constrained optimization, J. Numer. Math., 14 (2006), pp. 163185.
[4] Bium H. and Rannacher R., On mixed finite element methods in plate bending analysis, Comput. Mech., 6 (1990), pp. 221236.
[5] Brezzi F. and Fortin M., Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin, 1991.
[6] Casas E. and Fernández L. A., Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state, Appl. Math. Optim., 27 (1993), pp. 3556.
[7] Carstensen C., Gallistl D. and Hu J., A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles, Numer. Math., 124 (2013), pp. 309335.
[8] Cao W. and Yang D., Ciarlet-Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation, J. Comput. Appl. Math., 233 (2009), pp. 372388.
[9] Cheng X. L., Han W. M. and Huang H. C., Some mixed finite element methods for biharmonic equation, J. Comput. Appl. Math., 126 (2000), pp. 91109.
[10] Chen Y. and Hou T., Error estimates and superconvergence of RT0 mixed methods for a class of semilinear elliptic optimal control problems, Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 637656.
[11] Chen Y. and Sun C., Error estimates and superconvergence of mixed finite element methods for fourth order hyperbolic control problems, Appl. Math. Comput., 244 (2014), pp. 642653.
[12] Chen Y. and Lin Z., A posteriori error estimates of semidiscrete mixed finite element methods for parabolic optimal control problems, E. Asian J. Appl. Math., 5 (2015), pp. 85108.
[13] Chen Y. and Lu Z., Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problems, Comput. Methods Appl. Mech. Eng., 199 (2010), pp. 14151423.
[14] Chen Y. and Lu Z., High Efficient and Accuracy Numerical Methods for Optimal Control Problems, Beijing, Science Press, 2015.
[15] Chen Y., Yi N. and Liu W.B., A Legendre Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46 (2008), pp. 22542275.
[16] Douglas J. and Roberts J.E., Global estimates for mixed finite element methods for second-order elliptic equations, Math. Comp., 44 (1985), pp. 3952.
[17] Hinze M., A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), pp. 4561.
[18] Frei S., Rannacher R. and Wollner W., A priori error estimates for the finite element discretization of optimal distributed control problems governed by the bi-harmonic operator, Calcolo, 50 (2013), pp. 165193.
[19] Johnson C., On the convergence of a mixed finite-element method for plate bending problems, Numer. Math., 21 (1973), pp. 4362.
[20] Li B. J. and Liu S. Y., On gradient-type optimization method utilizing mixed finite element approximation for optimal boundary control problem governed by bi-harmonic equation, Appl. Math. Comput., 186 (2007), pp. 14291440.
[21] Li J., Optimal error estimates of mixed finite element methods for a fourth-order nonlinear elliptic problem, J. Math. Anal. Appl., 334 (2007), pp. 183195.
[22] Lions J.L., Optimal Control of Systems governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
[23] Lin J. and Lin Q., Supercovergence of a finite element method for the bi-harmonic equation, Numer. Methods Partial Differential Equations., 18 (2002), pp. 420427.
[24] Liu W. and Yan N., Adaptive Finite Element Methods for Optimal Control Governed by Partial Differential Equations, Science Press, Beijing, 2008.
[25] Lu Z. and Chen Y., L-error estimates of triangular mixed finite element methods for optimal control problem governed by semilinear elliptic equation, Numer. Anal. Appl., 12 (2009), pp. 7486.
[26] Kong L., On a fourth order elliptic problem with a p(x) bi-harmonic operator, Appl. Math. Lett., 27 (2014), pp. 2125.
[27] Monk P., A mixed finite element method for the bi-harmonic equation, SIAM J. Numer. Anal., 24 (1987), pp. 737749.
[28] Raviart P.A. and Thomas J.M., A mixed finite element method for 2nd order elliptic problems, In: Math. Aspects of the Finite Element Method, Lecture Notes in Math, 292-315, Springer-Verlag, Berlin, 1977.
[29] Scott R., A Mixed method for 4th order problems using linear finite elements, RARIO Anal. Numer., 33 (1978), pp. 681697.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 2
Total number of PDF views: 121 *
Loading metrics...

Abstract views

Total abstract views: 301 *
Loading metrics...

* Views captured on Cambridge Core between 17th November 2016 - 19th January 2018. This data will be updated every 24 hours.